Extremal problems on polynomials

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Paul Erdős, "Extremal problems on polynomials," in Approximation Theory II, pp. 347-355, Academic Press, 1976.

Markdown. source card.

Read status. The E1045 statement and historical report, and the E1150 formulation, were checked against the full paper in Markdown. This source gives no proof or construction for either cited problem.

The distance-product problem and its 1976 standing

In Section 3 (printed p. 350; local Markdown page 4), Erdős takes complex points satisfying

$$|z_i-z_j|\leq 2\qquad(1\leq i<j\leq n)$$

and asks whether

$$\prod_{1\leq i<j\leq n}|z_i-z_j|$$

is maximized by a regular polygon. This is the E1045 objective in unordered form: E1045 uses $\prod_{i\ne j}|z_i-z_j|$, the square of the displayed product, so the two normalizations have the same maximizing configurations.

The source says that Erdős, Herzog, and Piranian had made the conjecture in their earlier paper [7], Metric properties of polynomials (1958), and that Danzer and Pommerenke [3], Über die Diskriminante von Mengen gegebenen Durchmessers (1967), disproved it for even $n$. Erdős nevertheless writes that regular-polygon optimality "probably holds" for odd $n$ and, in that context, that the problem was open for $n\geq5$. Thus p. 350 is historical statement-and-status evidence: it records the original conjecture, the even-order disproof, and Erdős's surviving odd-order expectation as of 1976. It is not itself a proof of any of those assertions and does not establish the problem's modern status.

There is a normalization blemish in the printed sentence as represented by the reading copy: after imposing $|z_i-z_j|\leq2$, it calls the comparator a regular polygon "of diameter 1." Taken literally that polygon cannot maximize a positive homogeneous distance product, since scaling it to diameter $2$ strictly increases the product. The scale-consistent reading, and the one that matches E1045, is a regular polygon scaled to diameter $2$. The survey supplies neither the Danzer--Pommerenke counterconfiguration nor its product calculation; its mechanism and quantitative strength must therefore be obtained from the cited 1967 paper rather than inferred from this retrospective notice.

Relation to E1150

Section 8 (printed pp. 354–355) asks whether there is an absolute constant $c>0$ such that, for every choice of signs $\varepsilon_k\in\{-1,1\}$,

$$\max_{|z|=1}\left|\sum_{k=1}^{n}\varepsilon_k z^k\right| >(1+c)\sqrt n.$$

This is the E1150 conjecture in a shifted indexing convention: multiplying a degree-$N$ polynomial $P_N(z)=\sum_{j=0}^{N}\varepsilon_jz^j$ by $z$ gives the displayed sum with $n=N+1$ without changing its modulus on the unit circle. The paper supplies the formulation and historical context, but no proof, construction, or quantitative partial result for it. Its neighboring complex-unimodular version belongs to the larger coefficient class later shown by Kahane to admit ultraflat sequences; that does not settle the real $\{\pm1\}$ case posed in E1150.